6. Conclusions and Summary

This report has demonstrated many of the challenges in aggregating and disaggregating descriptions of combat processes by working through an analytically tractable model that assumes the Lanchester square law for ground combat in an individual sector such as that controlled by a corps or, in relatively rough terrain, a division. Even though this assumption is certainly not rigorous, it is nonetheless useful for the purposes here and leads to the following conclusions.

Aggregating from a Sector Level

Given a Lanchester law at the individual sector level, there may or may not be a valid aggregate model at the theater level. If the attacker and defender apply forces uniformly across sectors and maintain constant reserve fractions, an aggregate model exists and is itself Lanchesterian. If the attacker concentrates forces on a fraction of the sectors, conducting mere holding actions on the others, an aggregate model with constant coefficients is still valid so long as there is no change in the reserve fractions or the allocation of forces across sectors. Again the model is Lanchesterian. In this case, however, the key coefficient governing the ratio of loss rates is a complex function of the attacker's strategy and the defender's anticipation of the attacker's strategy (a function of information and decisionmaking). If the break-even force ratio at the sector level is 3:1, then the break-even force ratio at the theater level is about 1.5, 1.2, or 2.1 for canonical, defense-conservative, and attacker-conservative assumptions, respectively.

Effects of Intra-Battle Reinforcement

If in the course of battle the sides commit their reserves and redeploy forces from other sectors to the main sectors, there is no exact aggregate model with constant coefficients except in extreme cases. What matters are the ratios of several time scales: the duration of battle, the time to reinforce with theater reserves, and the time to redeploy from other sectors. The reinforcement and redeployment times depend not only on physical distances, roads, and movement rates, but also on intelligence, decision times, logistics, and the effects of air power. If, for example, the sector-level battle is intense enough or decisions slow enough, then the intra-battle maneuver and reinforcement will be too late and sector outcomes will depend on the initial sector- level force ratios, thereby favoring the attacker. By contrast, if the defender in a main sector can diagnose events quickly and control the pace of events, perhaps by virtue of multiple prepared lines or giving up space for time, then this will not be so and the value of the initial concentration will be less.

Break-Even Force Ratios at Different Scales (Different Levels of Combat)

While the importance of the relative time scales may seem obvious, these scales are seldom discussed explicitly, even though it has been mysterious to many observers over the years why the 3:1 rule is applied at some levels of combat but not others. As discussed above, if the 3:1 rule is valid at a sector level, the corresponding rule at a theater level may be more like 1.5:1. On the other hand, it is plausible for a 3:1 rule to apply not only at the sector level, but at the subsector level as well. Thus, the same rule might apply to battalion- and division-level battles. The general principle is that if a 3:1 rule applies at a given level, then it will also apply reasonably well at the next- higher level if the higher level's defensive resources can be reallocated in a much shorter time than the duration of lower- level battles (or if the attacker is unable to enforce concentration systematically). If defending forces can break off battles quickly, this increases the effective duration of the low- level battles, thereby allowing more time for "reequilibration." This gives "active" and "mobile" defense concepts advantages over purely static defenses, although static defenses can often exploit fortifications better.

Implications of Mobile Combat

In mobile warfare the defender has less advantage. In this case, the sector-level break-even force ratio is 1:1 and the break-even force ratio at the aggregate level may be on the order of 0.8--that is, even an outnumbered side can win. The risks of doing so are considerable, however, because holding actions are more difficult. Battles may be more intense and their durations correspondingly shorter. As a result, concentration of force can be decisive--again, unless defending commanders are deft at breaking off battle when outnumbered and maneuvering quickly to reinforce troubled units. Such maneuver issues are especially important today, because the United States is more likely than not to be engaging in mobile warfare rather than a rigid prepared defense of a fixed line.

Disaggregation and Reaggregation Within Combat Simulation Runs

Using the insights about aggregation relationships, it is possible to draw conclusions about temporary disaggregation in the course of a simulated battle (e.g., in a distributed simulation). By and large, disaggregating from a theater level in which the independent variables are total attacker and defender force levels is arbitrary and unnatural: It amounts to assuming a particular attack strategy for the entire campaign. Such an assumption cannot then be forgotten as one reaggregates, because in the real world theater-level strategies are highly correlated over time (i.e., if the main attack is through the Ardennes on D+1, the Ardennes is probably still a main sector on D+2). By contrast, it is not unreasonable to disaggregate temporarily from a sector-level description to a representative subsector-level depiction, and then reaggregate, if the time scales are such that one would expect forces in the sector to "reequilibrate" before the next time period requiring a disaggregated description.

Generic Principles

The purpose of the analysis is more to illustrate methods of aggregation and disaggregation than to work through the implications of the Lanchester square law. Among the more important principles illustrated are the following:

Even approximate mathematical analysis can clarify aggregation and disaggregation issues by suggesting functional forms and likely sensitivities.
However, aggregation typically depends sensitively on assumptions outside the detailed model, notably assumptions about higher-level strategy, command-control, maneuver, and time scales. These cannot generally be determined in advance, making uncertainty analysis necessary at the aggregate level.
The often dominating role of these higher-level factors is the reason that aggregate models (even board games) can often be quite respectable without being derived in detail from, or calibrated against, detailed models.
Aggregation may also depend sensitively on other assumptions outside the detailed model, assumptions so implicit as to be largely forgotten. The "detailed" models may, for example, be deterministic because of implicitly assumed tactics such as maintaining reserves that hedge against the consequences of random events. These assumptions must be reflected as constraints when aggregating or using automated methods such as neural nets or mathematical programming to find "optimal" tactics.
Temporary disaggregation within simulated campaigns may or may not be reasonable, depending on the objectives of the simulation and, importantly, the time scales involved. By and large, temporary disaggregation is defensible if, in the real world, forces would "reequilibrate" at the aggregate level between periods in which the simulation disaggregates. The reequilibration concept is general, not restricted to ground-force maneuver. The "reequilibration" may involve, e.g., alertness, allocation of fires, redeployment of command and control assets, or maneuver of aircraft and ships.
Validation of aggregation/disaggregation relationships should focus on the treatment of strategy, command-control, constraints, time scales, and uncertainties. It should not pivot around whether the aggregate model has been fully calibrated against a detailed model, because in many cases such calibration is impossible without mischievous assumptions. On the other hand, experiments with detailed models can often reveal issues and sensitivities that would be missed in even a moderately careful mathematical analysis. Further, they may be a good basis for calibrating some parameters of the aggregate model, even though other parameters are outside the model.

A corollary of the last point is that in developing families of models, it may be better to start with more aggregate concepts and develop consistent disaggregated representations and only partial calibrations than to attempt to work exclusively from the bottom up. This may be a radical concept to those wedded to bottom-up approaches. It is contrary to much current discussion, especially by some enthusiasts of distributed interactive simulation who happen to be more acquainted with training and distributed technology than with modeling.

At the same time, work with high-resolution models can be extremely important in clarifying underlying cause-effect relationships, defining the form of aggregate-level models, and calibrating specific parameters within them. As in this report, it is important to work from both directions and to fully appreciate what each level's perspective brings to the problem.

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